We have proved the following formulas:

In each case we have a formula of the form

This is a general sort of situation, as is shown by the following theorem.

Proof: Choose a point and define

Then for any points in we have

We've used the fact that

A function that has an indefinite integral always has infinitely many
indefinite integrals,
since if is an indefinite integral for then so is
for any number :

The following notation is used for indefinite integrals. One writes to denote an indefinite integral for . The here is a dummy variable and can be replaced by any available symbol. Thus, based on formulas (9.59) - (9.60), we write

We might also write

Some books always include an arbitrary constant with indefinite integrals, e.g.,

The notation for indefinite integrals is treacherous. If you see the two equations

and

then you want to conclude

which is wrong. It would be more logical to let the symbol denote the set of

and

you are not tempted to make the conclusion in (9.63).

Proof: The statement (9.65) means that if is an indefinite integral for and is an indefinite integral for , then is an indefinite integral for .

Let be an indefinite integral for and let be an indefinite integral for . Then for all

It follows that is an indefinite integral for .

while

Sometimes we write instead of .

and this notation is used as follows:

In the last example I have implicitly used

We have

so

Thus

Hence

> int((sin(x))^3*cos(3*x),x=0..Pi/2);responds with the value

- 5/12

Then determine the values of

without doing any calculations. (But include an explanation of where your answer comes from.)

Arrange the numbers in increasing order. Try to do the problem without making any explicit calculations. By making rough sketches of the graphs you should be able to come up with the answers. Sketch the graphs, and explain how you arrived at your conclusion. No `` proof" is needed.